Actually I was reading about permutation glass, it’s a about a disordered system where the state space consists of various orderings of a list and it’s based on permutation graph (please have a quick look at it, and with more attention at *pages 3-5, eq. 12, 14, 17*).

My interest is that I’am trying to apply the advanced probability model of random permutuation and to see if any contribution exists (from the side of math). The idea of probability model is that Gaussian is the function of two variables and it tends to the normal distribution for infinite numbers of elements. $ F_N (x)=Φ(x)+\frac{3}{50\sqrt{2π}} e^{-x^2/2} (x^3-3x) \frac {6N^3+21N^2+31N+31} {N(2N+5)^2 (N-1)}+O(\frac{1}{N^2} )$

Acording to the article it is supposed that Gaussian does not have the cumulants of order higher than $ 2$ . This results in more or less easy calculation (eq. 12, 14, 17).

However, there are higher cumulants greater then two (only odd cumulants equal to zero). In other words it was considered a distribution like below: $ \rho_0(\lambda)=\frac {1} {\sqrt{2\pi\sigma_0^2}}exp(-\frac{(\lambda-\lambda_0)^2} {2\sigma_0^2})$ .

I would like to consider Gaussian which has higher cumulants: $ \rho_0(\lambda)=\frac {1} {\sqrt{2\pi\sigma_0^2}}exp(-\frac{(\lambda-\lambda_0)^2} {2\sigma_0^2})+\frac {3} {50\sqrt{2\pi\sigma_0^2}}\frac {6N^3+21N^2+31N+3} {N(2N+5)^2(N-1)}exp(-\frac{(\lambda-\lambda_0)^2} {2\sigma_0^2})(-(\frac{(\lambda-\lambda_0)^2} {2\sigma_0^2})^2+6\frac{(\lambda-\lambda_0)^2} {2\sigma_0^2}-3)+O(\frac{1}{N^2})$

So I wonder if could anybody help with the solution for eq.12 from the article to see if the contribution exists or not?